Export every object into separate files
Here's a snippet of code that exports FBX.  I tested it briefly and it appears to work.  It just iterates the list of all objects, selects each one individually, and calls the FBX exporter on the selected object.

import bpy

def export_all_fbx(exportFolder):
    objects = bpy.data.objects
    for object in objects:
        bpy.ops.object.select_all(action='DESELECT')
        object.select_set(state=True)
        exportName = exportFolder + object.name + '.fbx'
        bpy.ops.export_scene.fbx(filepath=exportName, use_selection=True)


Example usage, shown here with a Windows-style path.  Note that a trailing slash of some kind must be included due to use of simple concatenation of path and filename.  Also note that backslashes must be escaped (doubled) but forward-slashes do not.  Give it the name of an existing, empty folder with the trailing slash included.

export_all_fbx('C:\\outputFolder\\')